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A class of inflationary scenarios for primordial black hole (PBH) formationinclude a small barrier in the slope of the potential. There, the inflatonslows down, generating an enhancement of primordial perturbations. Moreover,the background solution overcomes the barrier at a very low speed, and largebackward quantum fluctuations can prevent certain regions from overshooting thebarrier. This leads to localized bubbles where the field remains trapped behindthe barrier. In such models, therefore, we have two distinct channels for PBHproduction: the standard adiabatic density perturbation channel and the bubblechannel. Here, we perform numerical simulations of bubble formation, addressingthe issues of initial conditions, critical amplitude and bubble expansion.Further, we explore the scaling behaviour of the co-moving size of bubbles withthe initial amplitude of the field fluctuation. We find that for small tomoderate non-Gaussianity $f_{\rm NL}\lesssim 2.6$, the threshold for theformation of vacuum bubbles agrees with previous analytical estimatesarXiv:1908.11357 to $5\%$ accuracy or so. We also show that the massdistribution for the two channels is different, leading to a slightly broaderrange of PBH masses. The bubble channel is subdominant for small $f_{\rm NL}$,and becomes dominant for $f_{\rm NL}\gtrsim 2.6$. We find that the mass of PBHsin the bubble channel is determined by an adiabatic overdensity surrounding thebubble at the end of inflation. Remarkably, the profile of this overdensityturns out to be of type-II. This represents a first clear example showing thatoverdensities of type-II can be dominant relative to the standard type-I. Wealso comment on exponential tails and on the fact that in models with localtype non-Gaussianity (such as the one considered here), the occurrence ofalternative channels can easily be inferred from unitarity considerations.

Formation of trapped vacuum bubbles during inflation, and consequences for PBH scenarios